Causal discovery is essential for understanding relationships among variables of interest in many scientific domains. In this paper, we focus on permutation-based methods for learning causal graphs in Linear Gaussian Acyclic Models (LiGAMs), where the permutation encodes a causal ordering of the variables. Existing methods in this setting do not scale due to their high computational complexity.

Proposition 1. Assume the robust loss function is non-decreasing in [0, 1 ], (0) = 0 and The proof of Proposition 1 is mainly adapted from the proof in Kim & Scott ( 2012). For any given function: R! We first introduce a few notations that are useful for stating this result. B> (2 +) |O| where is the failure probability. By adapting Lemma 1 in Nguyen et al. ( 2022c) to uniform concentration bound, ImageNet We use the full ImageNet dataset that contains 1 .

ML models call for an equitable model valuation method to price them. In particular, we investigate the black-box access setting which allows querying a model (to observe predictions) without disclosing model-specific information (e.g., architecture and parameters). By exploiting a Dirichlet abstraction of a model's predictions, we propose a novel and equitable model valuation method called

Probabilistic solvers provide a flexible and efficient framework for simulation, uncertainty quantification, and inference in dynamical systems.